Correlation & Correlation & RegressionRegression

CorrelationCorrelation

Finding the relationship between two quantitative variables without being able to infer causal relationships

Correlation is a statistical technique used to determine the degree to which two variables are related

• Rectangular coordinate

• Two quantitative variables

• One variable is called independent (X) and

the second is called dependent (Y)

Scatter diagram

Example

Scatter diagram of weight and systolic blood Scatter diagram of weight and systolic blood pressurepressure

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SBP(mmHg)

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SBP(mmHg)

Scatter diagram of weight and systolic blood pressure

Scatter plots

The pattern of data is indicative of the type of relationship between your two variables:

positive relationship negative relationship no relationship

Positive relationshipPositive relationship

0

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4

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18

0 10 20 30 40 50 60 70 80 90

Age in Weeks

Hei

gh

t in

CM

Negative relationshipNegative relationship

Reliability

Age of Car

No relationNo relation

Correlation CoefficientCorrelation Coefficient

Statistic showing the degree of relation between two variables

Simple Correlation coefficient Simple Correlation coefficient (r)(r)

It is also called Pearson's correlation It is also called Pearson's correlation or product moment correlation or product moment correlationcoefficient. coefficient.

It measures the It measures the naturenature and and strengthstrength between two variables ofbetween two variables ofthe the quantitativequantitative type. type.

The The signsign of of rr denotes the nature of denotes the nature of association association

while the while the valuevalue of of rr denotes the denotes the strength of association.strength of association.

If the sign is If the sign is +ve+ve this means the relation this means the relation is is direct direct (an increase in one variable is (an increase in one variable is associated with an increase in theassociated with an increase in theother variable and a decrease in one other variable and a decrease in one variable is associated with avariable is associated with adecrease in the other variable).decrease in the other variable).

While if the sign is While if the sign is -ve-ve this means an this means an inverse or indirectinverse or indirect relationship (which relationship (which means an increase in one variable is means an increase in one variable is associated with a decrease in the other).associated with a decrease in the other).

The value of r ranges between ( -1) and ( +1)The value of r ranges between ( -1) and ( +1) The value of r denotes the strength of the The value of r denotes the strength of the

association as illustratedassociation as illustratedby the following diagram.by the following diagram.

-1 10-0.25-0.75 0.750.25

strong strongintermediate intermediateweak weak

no relation

perfect correlation

perfect correlation

Directindirect

If If rr = Zero = Zero this means no association or this means no association or correlation between the two variables.correlation between the two variables.

If If 0 < 0 < rr < 0.25 < 0.25 = weak correlation. = weak correlation.

If If 0.25 ≤ 0.25 ≤ rr < 0.75 < 0.75 = intermediate correlation. = intermediate correlation.

If If 0.75 ≤ 0.75 ≤ rr < 1 < 1 = strong correlation. = strong correlation.

If If r r = l= l = perfect correlation. = perfect correlation.

n

y)(y.

n

x)(x

n

yxxy

r2

22

2

How to compute the simple correlation coefficient (r)

ExampleExample::

A sample of 6 children was selected, data about their A sample of 6 children was selected, data about their age in years and weight in kilograms was recorded as age in years and weight in kilograms was recorded as shown in the following table . It is required to find the shown in the following table . It is required to find the correlation between age and weight.correlation between age and weight.

serial No

Age (years)

Weight (Kg)

1712

268

3812

4510

5611

6913

These 2 variables are of the quantitative type, one These 2 variables are of the quantitative type, one variable (Age) is called the independent and variable (Age) is called the independent and denoted as (X) variable and the other (weight)denoted as (X) variable and the other (weight)is called the dependent and denoted as (Y) is called the dependent and denoted as (Y) variables to find the relation between age and variables to find the relation between age and weight compute the simple correlation coefficient weight compute the simple correlation coefficient using the following formula:using the following formula:

n

y)(y.

n

x)(x

n

yxxy

r2

22

2

Serial n.

Age (years)

(x)

Weight (Kg)

(y)xyX2Y2

17128449144

268483664

38129664144

45105025100

56116636121

691311781169

Total∑x=41

∑y=66

∑xy= 461

∑x2=291

∑y2=742

r = 0.759r = 0.759

strong direct correlation strong direct correlation

6

(66)742.

6

(41)291

6

6641461

r22

EXAMPLE: Relationship between Anxiety and EXAMPLE: Relationship between Anxiety and Test ScoresTest Scores

AnxietyAnxiety

))XX((

Test Test score (Y)score (Y)

XX22YY22XYXY

101022100100442020

88336464992424

22994481811818

117711494977

5566252536363030

6655363625253030

∑∑X = 32X = 32∑∑Y = 32Y = 32∑∑XX22 = 230 = 230∑∑YY22 = 204 = 204∑∑XY=129XY=129

Calculating Correlation CoefficientCalculating Correlation Coefficient

94.)200)(356(

1024774

32)204(632)230(6

)32)(32()129)(6(22

r

r = - 0.94

Indirect strong correlation

Regression AnalysesRegression Analyses

Regression: technique concerned with predicting some variables by knowing others

The process of predicting variable Y using variable X

RegressionRegression

Uses a variable (x) to predict some outcome Uses a variable (x) to predict some outcome variable (y)variable (y)

Tells you how values in y change as a function Tells you how values in y change as a function of changes in values of xof changes in values of x

Correlation and RegressionCorrelation and Regression

Correlation describes the strength of a Correlation describes the strength of a linear relationship between two variables

Linear means “straight line”

Regression tells us how to draw the straight line described by the correlation

Regression Calculates the “best-fit” line for a certain set of dataCalculates the “best-fit” line for a certain set of data

The regression line makes the sum of the squares of The regression line makes the sum of the squares of the residuals smaller than for any other linethe residuals smaller than for any other line

Regression minimizes residuals

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By using the least squares method (a procedure By using the least squares method (a procedure that minimizes the vertical deviations of plotted that minimizes the vertical deviations of plotted points surrounding a straight line) we arepoints surrounding a straight line) we areable to construct a best fitting straight line to the able to construct a best fitting straight line to the scatter diagram points and then formulate a scatter diagram points and then formulate a regression equation in the form of:regression equation in the form of:

n

x)(x

n

yxxy

b2

2

1)xb(xyy b

bXay

Regression Equation

Regression equation describes the regression line mathematically Intercept Slope 80

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SBP(mmHg)

Linear EquationsLinear EquationsLinear EquationsLinear Equations

Y

Y = bX + a

a = Y-intercept

X

Changein Y

Change in X

b = Slope

bXay

Hours studying and Hours studying and gradesgrades

Regressing grades on hours grades on hours

Linear Regression

2.00 4.00 6.00 8.00 10.00

Number of hours spent studying

70.00

80.00

90.00

Final grade in course = 59.95 + 3.17 * studyR-Square = 0.88

Predicted final grade in class =

59.95 + 3.17*(number of hours you study per week)

Predict the final grade ofPredict the final grade of……

Someone who studies for 12 hours Final grade = 59.95 + (3.17*12) Final grade = 97.99

Someone who studies for 1 hour: Final grade = 59.95 + (3.17*1) Final grade = 63.12

Predicted final grade in class = 59.95 + 3.17*(hours of study)

Multiple Regression

Multiple regression analysis is a straightforward extension of simple regression analysis which allows more than one independent variable.